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Overview: This course is a graduate-level introduction to Algebraic and Enumerative Combinatorics. It is meant to cover most of the Module#1 material for the Oral Examination in Combinatorics administered at UPenn Mathematics Department. Some buzzwords: permutations, descents, q-analogues, generating functions, partition identities, bijections, posets.
Textbook: Enumerative Combinatorics Volume 1 (aka EC1), by Richard Stanley. Unless otherwise stated, we will follow the new second edition, available online (Note also the Errata link ther, as errors are still found.)
Schedule and grading:
There will be two in-class exams, one on Monday November 3 and one final exam on Wednesday December 10.
The class will also meet on 3 Fridays, dates and times to be agreed with the enrolled students.
Syllabus: Chapters 1,2, 3.1-3.8 of EC1 and further material:
Day-to-day schedule and homework problems are available here.
Chapter 1. What is Enumerative Combinatorics? 1.1 How to count 1.2 Sets and multisets 1.3 Cycles and inversions 1.4 Descents 1.5 Geometric representations of permutations 1.6 Alternating permutations, Euler numbers, and the cd-index of Sn 1.6.1 Basic properties 1.6.2 Flip equivalence of increasing binary trees 1.6.3 Min-max trees and the cd-index 1.7 Permutations of multisets 1.8 Partition identities 1.9 The Twelvefold Way 1.10 Two q-analogues of permutations 1.10.1 A q-analogue of permutations as bijections 1.10.2 A q-analogue of permutations as words Chapter 2. Sieve Methods 2.1 Inclusion-Exclusion 2.2 Examples and Special Cases 2.3 Permutations with Restricted Positions 2.4 Ferrers Boards 2.5 V -partitions and Unimodal Sequences 2.6 Involutions 2.7 Determinants Chapter 3. Partially Ordered Sets 3.1 Basic Concepts 3.2 New Posets from Old 3.3 Lattices 3.4 Distributive Lattices 3.5 Chains in Distributive Lattices 3.6 Incidence Algebras 3.7 The Möbius Inversion Formula 3.8 Techniques for Computing Möbius Functions